Magnitude homology is a homology theory of enriched categories, proposed by Michael Shulman late last year. For ordinary categories, it is the usual homology of a category (or equivalently, of its classifying space). But for metric spaces, regarded as enriched categories à la Lawvere, magnitude homology is something new. It gives truly metric information: for instance, the first homology of a subset X of R^n detects whether X is convex.

Like all homology theories, magnitude homology has an Euler characteristic, defined as the alternating sum of the ranks of the homology groups. Often this sum diverges, so we have to use some formal trickery to evaluate it. In this way, we end up with an Euler characteristic that is often not an integer. This number is called the “magnitude” of the enriched category. In topological settings it is the ordinary Euler characteristic, and in metric settings it is closely related to volume, surface area and other classical invariants of geometry.